Average Error: 0.2 → 0.1
Time: 19.7s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{6}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)} \cdot \left(x - 1\right)\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{6}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)} \cdot \left(x - 1\right)
double f(double x) {
        double r43498919 = 6.0;
        double r43498920 = x;
        double r43498921 = 1.0;
        double r43498922 = r43498920 - r43498921;
        double r43498923 = r43498919 * r43498922;
        double r43498924 = r43498920 + r43498921;
        double r43498925 = 4.0;
        double r43498926 = sqrt(r43498920);
        double r43498927 = r43498925 * r43498926;
        double r43498928 = r43498924 + r43498927;
        double r43498929 = r43498923 / r43498928;
        return r43498929;
}
↓
double f(double x) {
        double r43498930 = 6.0;
        double r43498931 = x;
        double r43498932 = sqrt(r43498931);
        double r43498933 = 4.0;
        double r43498934 = 1.0;
        double r43498935 = r43498931 + r43498934;
        double r43498936 = fma(r43498932, r43498933, r43498935);
        double r43498937 = r43498930 / r43498936;
        double r43498938 = r43498931 - r43498934;
        double r43498939 = r43498937 * r43498938;
        return r43498939;
}

Error

Bits error versus x

Target

Original0.2
Target0.1
Herbie0.1
\[\frac{6}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}}\]

Derivation

  1. Initial program 0.2

    \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\frac{6}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{x - 1}}}\]
  3. Using strategy rm

  4. Applied associate-/r/0.1

    \[\leadsto \color{blue}{\frac{6}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)} \cdot \left(x - 1\right)}\]

  5. Final simplification0.1

    \[\leadsto \frac{6}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)} \cdot \left(x - 1\right)\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"

  :herbie-target
  (/ 6.0 (/ (+ (+ x 1.0) (* 4.0 (sqrt x))) (- x 1.0)))

  (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))